Time variation of particle and antiparticle asymmetry in an expanding universe

Particle-number-violating interactions wash out the primordial asymmetry of the particle number density generated by some interaction satisfying Sakharov conditions for baryogenesis. In this paper, we study how the primordial asymmetry evolves in time under the presence of particle-number-violating interactions and in the environment of an expanding universe. We introduce a complex scalar model with particle-number-violating mass terms and calculate the time evolution of the particle number density with nonequilibrium quantum field theory. We show how the time evolution of the number density depends on parameters, including the chemical potential related to the particle number, temperature, the size of the particle-number-violating mass terms, and the expansion rate of the universe. The number density behaves differently depending upon whether the chemical potential is larger or smaller than the rest mass of the scalar particle. When the chemical potential is smaller than the mass, the interference among the contribution of oscillators with various momenta reduces the number density in addition to the dilution due to the expansion of the universe. In the opposite case, the oscillation of the particle number density lasts for a long time and the cancellation due to the interference does not occur.

Figure 1

The plot shows the time-dependent factor $\mathrm{V}(\eta ,{\eta }_0,k)$ for different values of momentum $k$, the expansion rate $H$, and the particle-number-violating mass $B$. The chemical potential and the temperature are fixed as $(\mu ,T)=(5,20)$. The dash-dotted line shows the result for the case $(B,H,k)=(3,0.1,5)$, the solid line shows the result for the case (3, 0.2, 5), and the thick solid line shows the result for the case (1, 0.1, 5). The dashed line shows the result for the case with $(B,H,k)=(3,0.1,20)$.

Figure 2

Momentum dependence of the distribution function $\mathrm{h}(k,\mu ,T)$. We choose the parameters $(\mu ,T)=(5,20)$.

Figure 3

Momentum dependence of the distribution function $\mathrm{h}(k,\mu ,T)$. We choose the parameters $(\mu ,T)=(11,20)$.

Figure 4

Curvature ($H$) effect on the time evolution of the current density. The dash-dotted line, the solid line, and the thick solid line show the cases for $H=0.2$, 0.1, and 0, respectively. $(B,\mu ,T)=(1,5,20)$ for all the lines.

Figure 5

Curvature ($H$) effect on the time evolution of the current density. The dash-dotted line, the solid line, and the thick solid line show the cases for $H=0.2$, 0.1, and 0, respectively. $(B,\mu ,T)=(3,5,20)$ for all the lines.

Figure 6

The mass-squared difference ($m_2^2-m_1^2=2B^2$) dependence of the current density. The solid line and the thick solid line show the cases $B=3$ and $B=1$, respectively. $(H,\mu ,T)=(0.1,5,20)$ for all the lines.

Figure 7

Dependence on the temperature $T$ of the time evolution of the current density. The dashed line and the solid line show the cases $T=10$ and $T=20$, respectively. $(B,H,\mu )=(3,0.1,5)$ for all the lines.

Figure 8

Dependence on the chemical potential $\mu$ of the time evolution of the current density. The dashed line, the solid line, the thick solid line show the cases $\mu =5$, 11, and 20, respectively. $(B,H,T)=(3,0.1,10)$ for all the lines.

Figure 9

We show the time dependence of the current densities normalized by their initial values. The case with $\mu >m_{\varphi }$ and the case with $\mu <m_{\varphi }$ are shown. The thin solid line shows the case with $\mu =20$ and the thick solid line shows the case with $\mu =5$. We choose $m_{\varphi }=10$ and $(B,H,T)=(3,0.1,10)$ for both cases. For comparison, the dashed line shows the time dependence for the inverse of the universe’s volume, i.e., $e^{-3H{x}^0}$ with $H=0.1$.

Figure 10

The thick solid line shows the contour $R$.